Optimal. Leaf size=138 \[ -\frac {a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}-\frac {\left (d h^2-e g h+f g^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{\left (a h^2+c g^2\right )^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1647, 12, 725, 206} \[ -\frac {a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}-\frac {\left (d h^2-e g h+f g^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{\left (a h^2+c g^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 725
Rule 1647
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2}{(g+h x) \left (a+c x^2\right )^{3/2}} \, dx &=-\frac {a (c e g-c d h+a f h)-c (c d g-a f g+a e h) x}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}+\frac {\int \frac {a c \left (f g^2-e g h+d h^2\right )}{\left (c g^2+a h^2\right ) (g+h x) \sqrt {a+c x^2}} \, dx}{a c}\\ &=-\frac {a (c e g-c d h+a f h)-c (c d g-a f g+a e h) x}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}+\frac {\left (f g^2-e g h+d h^2\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{c g^2+a h^2}\\ &=-\frac {a (c e g-c d h+a f h)-c (c d g-a f g+a e h) x}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}-\frac {\left (f g^2-e g h+d h^2\right ) \operatorname {Subst}\left (\int \frac {1}{c g^2+a h^2-x^2} \, dx,x,\frac {a h-c g x}{\sqrt {a+c x^2}}\right )}{c g^2+a h^2}\\ &=-\frac {a (c e g-c d h+a f h)-c (c d g-a f g+a e h) x}{a c \left (c g^2+a h^2\right ) \sqrt {a+c x^2}}-\frac {\left (f g^2-e g h+d h^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{\left (c g^2+a h^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 137, normalized size = 0.99 \[ \frac {a^2 (-f) h+a c (d h-e g+e h x-f g x)+c^2 d g x}{a c \sqrt {a+c x^2} \left (a h^2+c g^2\right )}-\frac {\left (h (d h-e g)+f g^2\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{\left (a h^2+c g^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 4.00, size = 721, normalized size = 5.22 \[ \left [\frac {{\left (a^{2} c f g^{2} - a^{2} c e g h + a^{2} c d h^{2} + {\left (a c^{2} f g^{2} - a c^{2} e g h + a c^{2} d h^{2}\right )} x^{2}\right )} \sqrt {c g^{2} + a h^{2}} \log \left (\frac {2 \, a c g h x - a c g^{2} - 2 \, a^{2} h^{2} - {\left (2 \, c^{2} g^{2} + a c h^{2}\right )} x^{2} - 2 \, \sqrt {c g^{2} + a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{h^{2} x^{2} + 2 \, g h x + g^{2}}\right ) - 2 \, {\left (a c^{2} e g^{3} + a^{2} c e g h^{2} - {\left (a c^{2} d - a^{2} c f\right )} g^{2} h - {\left (a^{2} c d - a^{3} f\right )} h^{3} - {\left (a c^{2} e g^{2} h + a^{2} c e h^{3} + {\left (c^{3} d - a c^{2} f\right )} g^{3} + {\left (a c^{2} d - a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} g^{4} + 2 \, a^{3} c^{2} g^{2} h^{2} + a^{4} c h^{4} + {\left (a c^{4} g^{4} + 2 \, a^{2} c^{3} g^{2} h^{2} + a^{3} c^{2} h^{4}\right )} x^{2}\right )}}, -\frac {{\left (a^{2} c f g^{2} - a^{2} c e g h + a^{2} c d h^{2} + {\left (a c^{2} f g^{2} - a c^{2} e g h + a c^{2} d h^{2}\right )} x^{2}\right )} \sqrt {-c g^{2} - a h^{2}} \arctan \left (\frac {\sqrt {-c g^{2} - a h^{2}} {\left (c g x - a h\right )} \sqrt {c x^{2} + a}}{a c g^{2} + a^{2} h^{2} + {\left (c^{2} g^{2} + a c h^{2}\right )} x^{2}}\right ) + {\left (a c^{2} e g^{3} + a^{2} c e g h^{2} - {\left (a c^{2} d - a^{2} c f\right )} g^{2} h - {\left (a^{2} c d - a^{3} f\right )} h^{3} - {\left (a c^{2} e g^{2} h + a^{2} c e h^{3} + {\left (c^{3} d - a c^{2} f\right )} g^{3} + {\left (a c^{2} d - a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{a^{2} c^{3} g^{4} + 2 \, a^{3} c^{2} g^{2} h^{2} + a^{4} c h^{4} + {\left (a c^{4} g^{4} + 2 \, a^{2} c^{3} g^{2} h^{2} + a^{3} c^{2} h^{4}\right )} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 294, normalized size = 2.13 \[ \frac {\frac {{\left (c^{3} d g^{3} - a c^{2} f g^{3} + a c^{2} d g h^{2} - a^{2} c f g h^{2} + a c^{2} g^{2} h e + a^{2} c h^{3} e\right )} x}{a c^{3} g^{4} + 2 \, a^{2} c^{2} g^{2} h^{2} + a^{3} c h^{4}} + \frac {a c^{2} d g^{2} h - a^{2} c f g^{2} h + a^{2} c d h^{3} - a^{3} f h^{3} - a c^{2} g^{3} e - a^{2} c g h^{2} e}{a c^{3} g^{4} + 2 \, a^{2} c^{2} g^{2} h^{2} + a^{3} c h^{4}}}{\sqrt {c x^{2} + a}} - \frac {2 \, {\left (f g^{2} + d h^{2} - g h e\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} h + \sqrt {c} g}{\sqrt {-c g^{2} - a h^{2}}}\right )}{{\left (c g^{2} + a h^{2}\right )} \sqrt {-c g^{2} - a h^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 862, normalized size = 6.25 \[ \frac {c d g x}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, a}-\frac {c e \,g^{2} x}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, a h}+\frac {c f \,g^{3} x}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, a \,h^{2}}-\frac {d h \ln \left (\frac {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {e g \ln \left (\frac {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}-\frac {f \,g^{2} \ln \left (\frac {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, h}+\frac {d h}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}-\frac {e g}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {f \,g^{2}}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {g}{h}\right ) c g}{h}+\left (x +\frac {g}{h}\right )^{2} c +\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, h}+\frac {e x}{\sqrt {c \,x^{2}+a}\, a h}-\frac {f g x}{\sqrt {c \,x^{2}+a}\, a \,h^{2}}-\frac {f}{\sqrt {c \,x^{2}+a}\, c h} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 453, normalized size = 3.28 \[ \frac {c f g^{3} x}{\sqrt {c x^{2} + a} a c g^{2} h^{2} + \sqrt {c x^{2} + a} a^{2} h^{4}} - \frac {c e g^{2} x}{\sqrt {c x^{2} + a} a c g^{2} h + \sqrt {c x^{2} + a} a^{2} h^{3}} + \frac {c d g x}{\sqrt {c x^{2} + a} a c g^{2} + \sqrt {c x^{2} + a} a^{2} h^{2}} + \frac {f g^{2}}{\sqrt {c x^{2} + a} c g^{2} h + \sqrt {c x^{2} + a} a h^{3}} - \frac {e g}{\sqrt {c x^{2} + a} c g^{2} + \sqrt {c x^{2} + a} a h^{2}} + \frac {d}{\frac {\sqrt {c x^{2} + a} c g^{2}}{h} + \sqrt {c x^{2} + a} a h} - \frac {f g x}{\sqrt {c x^{2} + a} a h^{2}} + \frac {e x}{\sqrt {c x^{2} + a} a h} + \frac {f g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{3}} - \frac {e g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h^{2}} + \frac {d \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} h} - \frac {f}{\sqrt {c x^{2} + a} c h} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {f\,x^2+e\,x+d}{\left (g+h\,x\right )\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d + e x + f x^{2}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (g + h x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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